Quantitative analysis of EEG signals: Time-frequency methods and ...

the sum of the positive exponents (i.e. the ones giving the rate of expansion of the
volume), equals the Kolmogorov entropy (Pesin, 1977).
K 2 = X >0
i (47)
5.2.6 Calculating Lyapunov Exponents
Wolf Method
Wolfetal. (1985) proposed an algorithm for calculating the largest Lyapunov exponent.
First, the phase space reconstruction is made (eq.43) and the nearest neighbor is
searched for one of the rst embedding vectors. A restriction must be made when searching
for the neighbor: it must be suciently separated in time in order not to compute
as nearest neighbors successive vectors of the same trajectory. Without considering this
correction, Lyapunov exponents could be spurious due to temporal correlation of the
neighbors. Once the neighbor and the initial distance (L) is determined, the system is
evolved forward some xed time (evolution time) and the new distance (L 0 ) is calculated.
This evolution is repeated, calculating the successive distances, until the separation is
greater than a certain threshold. Then a new vector (replacement vector) is searched as
close as possible to the rst one, having approximately the same orientation of the rst
neighbor. Finally, Lyapunov exponents can be estimated using the following formula:
L 1 =
1
(t k ; t 0 )
kX
i=1
ln L0 (t i )
L(t i;1 )
(48)
where k is the number of time propagation steps.
Rosenstein Method
Rosenstein et al. (1993) developed another algorithm for the calculation of the largest
Lyapunov Exponent in short and noisy time series. As before, the rst step is to make
a phase space reconstruction. Then the nearest neighbor for each embedding vector
is found. After this, the system is evolved some xed time and the largest Lyapunov
Exponent can be estimated as the mean rate of separation of the neighbors.
Assuming that the separation is determined by the largest Lyapunov Exponent (),
then at a time t the distance will be:
d(t) C e t (49)
where C is the initial separation. Taking the natural logarithm of both sides we obtain:
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